{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T07:52:09Z","timestamp":1771573929572,"version":"3.50.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2020,9,24]],"date-time":"2020-09-24T00:00:00Z","timestamp":1600905600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,9,24]],"date-time":"2020-09-24T00:00:00Z","timestamp":1600905600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100005758","name":"Universit\u00e0 Politecnica delle Marche","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100005758","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2021,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We show that the algebraic boundaries of the regions of real binary forms with fixed typical rank are always unions of dual varieties to suitable coincident root loci.\n<\/jats:p>","DOI":"10.1007\/s10208-020-09474-9","type":"journal-article","created":{"date-parts":[[2020,9,24]],"date-time":"2020-09-24T20:02:46Z","timestamp":1600977766000},"page":"1003-1022","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Algebraic Boundaries Among Typical Ranks for Real Binary Forms of Arbitrary Degree"],"prefix":"10.1007","volume":"21","author":[{"given":"Maria Chiara","family":"Brambilla","sequence":"first","affiliation":[]},{"given":"Giovanni","family":"Staglian\u00f2","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,9,24]]},"reference":[{"issue":"2","key":"9474_CR1","first-page":"201","volume":"4","author":"J Alexander","year":"1995","unstructured":"J.\u00a0Alexander and A.\u00a0Hirschowitz, Polynomial interpolation in several variables, J. Algebraic Geom.\u00a04 (1995), no.\u00a02, 201\u2013222.","journal-title":"J. Algebraic Geom."},{"issue":"3","key":"9474_CR2","doi-asserted-by":"publisher","first-page":"793","DOI":"10.1007\/s10208-013-9174-8","volume":"15","author":"G Blekherman","year":"2015","unstructured":"G.\u00a0Blekherman, Typical real ranks of binary forms, Found. Comput. Math.\u00a015 (2015), no.\u00a03, 793\u2013798.","journal-title":"Found. Comput. Math."},{"issue":"5","key":"9474_CR3","doi-asserted-by":"publisher","first-page":"1229","DOI":"10.1016\/j.jpaa.2007.09.014","volume":"212","author":"MC Brambilla","year":"2008","unstructured":"M.\u00a0C. Brambilla and G.\u00a0Ottaviani, On the Alexander\u2013Hirschowitz theorem, J. Pure Appl. Algebra\u00a0212 (2008), no.\u00a05, 1229\u20131251.","journal-title":"J. Pure Appl. Algebra"},{"key":"9474_CR4","unstructured":"M.\u00a0C. Brambilla and G.\u00a0Staglian\u00f2, CoincidentRootLoci: a macaulay2 package for computations with coincident root loci, version 0.1.2, online documentation and source code are available at https:\/\/faculty.math.illinois.edu\/Macaulay2\/doc\/Macaulay2-1.15\/share\/doc\/Macaulay2\/CoincidentRootLoci\/html, 2018."},{"key":"9474_CR5","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1016\/j.laa.2018.07.036","volume":"557","author":"MC Brambilla","year":"2018","unstructured":"M.\u00a0C. Brambilla and G.\u00a0Staglian\u00f2, On the algebraic boundaries among typical ranks for real binary forms, Linear Algebra Appl.\u00a0557 (2018), 403\u2013418.","journal-title":"Linear Algebra Appl."},{"issue":"1","key":"9474_CR6","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1007\/s10231-010-0137-2","volume":"190","author":"A Causa","year":"2011","unstructured":"A.\u00a0Causa and R.\u00a0Re, On the maximum rank of a real binary form, Ann. Mat. Pura Appl.\u00a0190 (2011), no.\u00a01, 55\u201359.","journal-title":"Ann. Mat. Pura Appl."},{"issue":"1","key":"9474_CR7","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1016\/S0021-8693(03)00336-3","volume":"267","author":"JV Chipalkatti","year":"2003","unstructured":"J.\u00a0V. Chipalkatti, On equations defining coincident root loci, J. Algebra\u00a0267 (2003), no.\u00a01, 246\u2013271.","journal-title":"J. Algebra"},{"key":"9474_CR8","doi-asserted-by":"crossref","unstructured":"J.\u00a0V. Chipalkatti, Invariant equations defining coincident root loci, Arch. Math. 83 (2004), no.\u00a05, 422\u2013428.","DOI":"10.1007\/s00013-004-1191-z"},{"issue":"1","key":"9474_CR9","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/s10208-010-9077-x","volume":"11","author":"G Comas","year":"2010","unstructured":"G.\u00a0Comas and M.\u00a0Seiguer, On the rank of a binary form, Found. Comput. Math.\u00a011 (2010), no.\u00a01, 65\u201378.","journal-title":"Found. Comput. Math."},{"issue":"3","key":"9474_CR10","doi-asserted-by":"publisher","first-page":"1254","DOI":"10.1137\/060661569","volume":"30","author":"P Comon","year":"2008","unstructured":"P.\u00a0Comon, G.\u00a0Golub, L.-H. Lim, and B.\u00a0Mourrain, Symmetric tensors and symmetric tensor rank, SIAM J. Matrix Anal. Appl. 30 (2008), no.\u00a03, 1254\u20131279.","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"2","key":"9474_CR11","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/0165-1684(96)00079-5","volume":"53","author":"P Comon","year":"1996","unstructured":"P.\u00a0Comon and B.\u00a0Mourrain, Decomposition of quantics in sums of powers of linear forms, Signal Processing 53 (1996), no.\u00a02, 93 \u2013 107.","journal-title":"Signal Processing"},{"issue":"6","key":"9474_CR12","doi-asserted-by":"publisher","first-page":"657","DOI":"10.1080\/03081087.2011.624097","volume":"60","author":"P Comon","year":"2012","unstructured":"P.\u00a0Comon and G.\u00a0Ottaviani, On the typical rank of real binary forms, Linear Multilinear Algebra 60 (2012), no.\u00a06, 657\u2013667.","journal-title":"Linear Multilinear Algebra"},{"issue":"3","key":"9474_CR13","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1006\/eujc.1993.1022","volume":"14","author":"R Ehrenborg","year":"1993","unstructured":"R.\u00a0Ehrenborg and G.-C. Rota, Apolarity and canonical forms for homogeneous polynomials, European J. Combin. 14 (1993), no.\u00a03, 157\u2013181.","journal-title":"European J. Combin."},{"key":"9474_CR14","doi-asserted-by":"crossref","unstructured":"I.\u00a0M. Gelfand, M.\u00a0M. Kapranov, and A.\u00a0V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, reprint of the 1994 edition ed., Birkh\u00e4user Boston, 2008.","DOI":"10.1007\/978-0-8176-4771-1"},{"key":"9474_CR15","unstructured":"D.\u00a0R. Grayson and M.\u00a0E. Stillman,Macaulay2\u2013 A software system for research in algebraic geometry (version 1.16), home page: http:\/\/www.math.uiuc.edu\/Macaulay2\/, 2020."},{"key":"9474_CR16","unstructured":"P.\u00a0Griffiths and J.\u00a0Harris, Principles of algebraic geometry, Pure Appl. Math., Wiley-Intersci., New York, 1978."},{"key":"9474_CR17","volume-title":"Differential topology","author":"V Guillemin","year":"2010","unstructured":"V.\u00a0Guillemin and A.\u00a0Pollack, Differential topology, American Mathematical Society, 2010."},{"key":"9474_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-28027-6","volume-title":"Tensor spaces and numerical tensor calculus","author":"W Hackbusch","year":"2012","unstructured":"W.\u00a0Hackbusch, Tensor spaces and numerical tensor calculus, Springer Berlin Heidelberg, 2012."},{"key":"9474_CR19","doi-asserted-by":"crossref","unstructured":"J.\u00a0Harris, Algebraic geometry: A first course, Grad. Texts in Math., vol. 133, Springer-Verlag, New York, 1992.","DOI":"10.1007\/978-1-4757-2189-8"},{"issue":"4","key":"9474_CR20","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1007\/BF01444088","volume":"30","author":"D Hilbert","year":"1887","unstructured":"D.\u00a0Hilbert, \u00dcber die Singularit\u00e4ten der Diskriminantenfl\u00e4che, Math. Ann. 30 (1887), no.\u00a04, 437\u2013441.","journal-title":"Math. Ann."},{"issue":"2","key":"9474_CR21","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/BF01259325","volume":"61","author":"A Holme","year":"1988","unstructured":"A.\u00a0Holme, The geometric and numerical properties of duality in projective algebraic geometry, Manuscripta Math. 61 (1988), no.\u00a02, 145\u2013162.","journal-title":"Manuscripta Math."},{"key":"9474_CR22","doi-asserted-by":"crossref","unstructured":"A.\u00a0Iarrobino and V.\u00a0Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, vol. 1721, Springer-Verlag, 1999, with an Appendix by A. Iarrobino and S. L. Kleiman.","DOI":"10.1007\/BFb0093426"},{"issue":"3","key":"9474_CR23","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1016\/S0723-0869(03)80002-6","volume":"21","author":"G Katz","year":"2003","unstructured":"G.\u00a0Katz, How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties, Expo. Math. 21 (2003), no.\u00a03, 219\u2013261.","journal-title":"Expo. Math."},{"key":"9474_CR24","doi-asserted-by":"publisher","unstructured":"K.\u00a0Kohn, Coisotropic hypersurfaces in Grassmannians, J. Symbolic Comput. (2019), in press, https:\/\/doi.org\/10.1016\/j.jsc.2019.12.002.","DOI":"10.1016\/j.jsc.2019.12.002"},{"issue":"3","key":"9474_CR25","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1137\/07070111X","volume":"51","author":"TG Kolda","year":"2009","unstructured":"T.\u00a0G. Kolda and B.\u00a0W. Bader, Tensor decompositions and applications, SIAM Rev. 51 (2009), no.\u00a03, 455\u2013500.","journal-title":"SIAM Rev."},{"issue":"1","key":"9474_CR26","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1016\/j.jalgebra.2011.10.045","volume":"352","author":"S Kurmann","year":"2012","unstructured":"S.\u00a0Kurmann, Some remarks on equations defining coincident root loci, J. Algebra 352 (2012), no.\u00a01, 223\u2013231.","journal-title":"J. Algebra"},{"key":"9474_CR27","doi-asserted-by":"crossref","unstructured":"J.\u00a0M. Landsberg, Tensors: geometry and applications, vol. 128, American Mathematical Soc., 2012.","DOI":"10.1090\/gsm\/128"},{"issue":"11","key":"9474_CR28","doi-asserted-by":"publisher","first-page":"285","DOI":"10.4086\/toc.2015.v011a011","volume":"11","author":"JM Landsberg","year":"2015","unstructured":"J.\u00a0M. Landsberg and G.\u00a0Ottaviani, New lower bounds for the border rank of matrix multiplication, Theory Comput. 11 (2015), no.\u00a011, 285\u2013298.","journal-title":"Theory Comput."},{"key":"9474_CR29","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1016\/j.jalgebra.2015.08.029","volume":"446","author":"H Lee","year":"2016","unstructured":"H.\u00a0Lee and B.\u00a0Sturmfels, Duality of multiple root loci, J. Algebra 446 (2016), 499\u2013526.","journal-title":"J. Algebra"},{"issue":"3","key":"9474_CR30","doi-asserted-by":"publisher","first-page":"1025","DOI":"10.1007\/s10231-016-0606-3","volume":"196","author":"M Micha\u0142ek","year":"2017","unstructured":"M.\u00a0Micha\u0142ek, H.\u00a0Moon, B.\u00a0Sturmfels, and E.\u00a0Ventura, Real rank geometry of ternary forms, Ann. Mat. Pura Appl. 196 (2017), no.\u00a03, 1025\u20131054.","journal-title":"Ann. Mat. Pura Appl."},{"issue":"3","key":"9474_CR31","doi-asserted-by":"publisher","first-page":"1308","DOI":"10.1016\/j.aim.2012.06.023","volume":"231","author":"L Oeding","year":"2012","unstructured":"L.\u00a0Oeding, Hyperdeterminants of polynomials, Adv. Math. 231 (2012), no.\u00a03, 1308\u20131326.","journal-title":"Adv. Math."},{"issue":"4","key":"9474_CR32","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1007\/s11464-007-0031-4","volume":"2","author":"L Qi","year":"2007","unstructured":"L.\u00a0Qi, W.\u00a0Sun, and Y.\u00a0Wang, Numerical multilinear algebra and its applications, Front. Math. China 2 (2007), no.\u00a04, 501\u2013526.","journal-title":"Front. Math. China"},{"key":"9474_CR33","doi-asserted-by":"crossref","unstructured":"A.\u00a0Smilde, R.\u00a0Bro, and P.\u00a0Geladi, Multi-way analysis with applications in the chemical sciences, John Wiley & Sons, Ltd, 2004.","DOI":"10.1002\/0470012110"},{"issue":"1","key":"9474_CR34","doi-asserted-by":"publisher","first-page":"21","DOI":"10.2140\/jsag.2018.8.21","volume":"8","author":"G Staglian\u00f2","year":"2018","unstructured":"G.\u00a0Staglian\u00f2, A package for computations with classical resultants, J. Softw. Alg. Geom. 8 (2018), no.\u00a01, 21\u201330.","journal-title":"J. Softw. Alg. Geom."},{"key":"9474_CR35","doi-asserted-by":"publisher","first-page":"186","DOI":"10.1016\/j.jsc.2016.08.012","volume":"79","author":"B Sturmfels","year":"2017","unstructured":"B.\u00a0Sturmfels, The Hurwitz form of a projective variety, J. Symbolic Comput. 79 (2017), 186\u2013196.","journal-title":"J. Symbolic Comput."},{"key":"9474_CR36","unstructured":"J.\u00a0J. Sylvester, An essay on canonical forms, supplement to a sketch of a memoir on elimination, transformation and canonical forms, 1851, Paper 34 in vol. 1 (publ. 1904), pp.\u00a0203\u2013216."},{"issue":"7","key":"9474_CR37","doi-asserted-by":"publisher","first-page":"1404","DOI":"10.1080\/03081087.2018.1454395","volume":"67","author":"E Ventura","year":"2019","unstructured":"E.\u00a0Ventura, Real rank boundaries and loci of forms, Linear and Multilinear Algebra 67 (2019), no.\u00a07, 1404\u20131419.","journal-title":"Linear and Multilinear Algebra"},{"issue":"1","key":"9474_CR38","doi-asserted-by":"publisher","first-page":"244","DOI":"10.1016\/0021-8693(89)90248-2","volume":"122","author":"J Weyman","year":"1989","unstructured":"J.\u00a0Weyman, The equations of strata for binary forms, J. Algebra 122 (1989), no.\u00a01, 244\u2013249.","journal-title":"J. Algebra"}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-020-09474-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10208-020-09474-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-020-09474-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,24]],"date-time":"2021-09-24T06:45:03Z","timestamp":1632465903000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10208-020-09474-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,24]]},"references-count":38,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,8]]}},"alternative-id":["9474"],"URL":"https:\/\/doi.org\/10.1007\/s10208-020-09474-9","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,9,24]]},"assertion":[{"value":"8 November 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 July 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 September 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 September 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}